Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental Directions2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)3008110.22363/2413-3639-2021-67-4-755-765Research ArticleInjectivity and Nuclearity Properties for Real C*-AlgebrasRakhimovA. A.rakhimov@ktu.edu.trNurillaevM. E.nur11laev_muzaffar@mail.ruBoltaevKh. Kh.bkhabibzhan@mail.ruNational University of Uzbekistan named after M. UlugbekTashkent State Pedagogical University named after Nizami3012202167475576524012022Copyright © 2022, Contemporary Mathematics. Fundamental Directions2022<p style="text-align: justify;">In this paper, we study injective and nuclear real W*- and C*-algebras. The connection of these concepts with similar concepts of enveloping W*- and C*-algebras is considered. The equivalence of the concepts of injectivity and nuclearity for real C*-algebras is shown. As a consequence, nuclear real factors of types II<sub>1</sub>, II<sub>∞</sub>, III<sub>1, </sub>III<sub>0</sub> and III<sub>λ</sub> (0 λ1) are completely described.</p>[Аюпов Ш. А. Классификация инъективных JW-факторов// Функц. анализ и его прилож. - 1984. - 18, № 3. - C. 67-68.][Фон Нейман Дж. Обобщение математического аппарата квантовой механики методами абстрактной алгебры. I// Мат. сб. - 1936. -1, № 4. - С. 415-485.][Фон Нейман Дж. Избранные труды по функциональному анализу. I, II. - М.: Наука, 1987.][Ayupov Sh. А., Rakhimov А. А., Usmanov Sh. M. Jordan, Real and Lie structures in operator algebras. - Dordrecht: Kluwer Academic Publishers, 1997.][Ayupov Sh. A., Rakhimov A. A. Real W*-algebras, actions of groups and index theory for real factors. - Beau-Bassin: VDM Verlag Dr. Muller, 2010.¨][Connes A. Classification of injective facteurs// Ann. Math. - 1976. -104, № 1. - С. 73-115.][Dixmier J. Les algebres d’operateurs dans l’espace Hilbertien. - Paris: Gauthier-Villars, 1969.][Grothendieck A. Resume des results essentiels dans la theorie des produits tensoriels topologiques et des espaces nucleaires// Ann. Inst. Fourier (Grenoble). - 1954. -4. - С. 73-112.][Grothendieck A. Produits tensoriels topologiques et des espaces nucleaires// S´ eminaire N. Bourbaki. -´ 1954. -69. - С. 193-200.][Li B. R. Real operator algebras. - River Edge: World Scientific, 2003.][Murray F., von Neumann J. On rings of operators. I// Ann. Math. - 1936. -37. - С. 116-229.][Rakhimov А. А., Nurillaev M. E. On property of injectivity for real W*-algebras and JW-algebras// Positivity. - 2018. - 22. - С. 1345-1354.][Sakai S. C*-algebras and W*-algebras. - Berlin: Springer, 1971.][Stacey P. J. Real structure in unital separable simple C*-algebras with tracial rank zero and with a unique tracial state// New York J. Math. - 2006. -12. - С. 269-273.][Stormer E. Real structure in the hyperfinite factor// Duke Math. J. - 1980. -47, № 1. - С. 145-153.][Takesaki M. Theory of operator algebras. I, II, III. - Berlin: Springer, 1979.]